Computer Science 320

Parallel Image Generation

The Mandelbrot Set

The Mandelbrot Set

A set of points defined as follows:

given a point (x, y), compute a sequence of other points (ai, bi), i = 0, 1, 2, …

a0 = 0b0 = 0ai+1 = ai2 – bi2 + xbi+1 = 2 * ai * bi + y

If each point in (ai, bi) stays finite, (x, y) is in; otherwise, not in

The Mandelbrot Set

If each point in (ai, bi) stays finite, (x, y) is in; otherwise, not in

Can’t do an infinite # of points, so

if (ai, bi) ever exceeds a distance of 2 from the origin, it’s not in

or: if (ai2 + bi2 )1/2 > 2 for some i

The Mandelbrot Set

or: if (ai2 + bi2 )1/2 > 2 for some i

If we do 1,000 points, if i reaches this limit before the distance exceeds 2, we’ll call the point in, even if further tests might show it to be out

Program Resources

• Will use the Parallel Java Graphics (PJG) format, which is lossless and larger, but uses faster compression, than PNG format

• Will generate color values for each point and save these in a PJG file

Program Inputs• Image width and height

• Coordinates of the image center point

• Image resolution in pixels per unit

• Maximum number of iterations to test for membership

• Exponent in formula to calculate pixel hues

• Output file name

Pixel Matrix, Image File, Hue Table// Create image matrix to store results.matrix = new int [height] [width];image = new PJGColorImage (height, width, matrix);

// Create table of hues for different iteration counts.huetable = new int [maxiter+1];for (int i = 0; i < maxiter; ++ i){ huetable[i] = HSB.pack(/*hue*/ (float) Math.pow(((double)i) / ((double)maxiter), gamma),

/*sat*/ 1.0f, /*bri*/ 1.0f);

}huetable[maxiter] = HSB.pack (1.0f, 1.0f, 0.0f);

Optimizing Matrix Access// Compute all rows and columns.for (int r = 0; r < height; ++ r){ int[] matrix_r = matrix[r]; double y = ycenter + (yoffset - r) / resolution; for (int c = 0; c < width; ++ c){

double x = xcenter + (xoffset + c) / resolution;

Allows access to a cell with a single index operation

Iterate Until Convergence// Iterate until convergence.int i = 0;double aold = 0.0;double bold = 0.0;double a = 0.0;double b = 0.0;double zmagsqr = 0.0;while (i < maxiter && zmagsqr <= 4.0){ ++ i; a = aold * aold – bold * bold + x; b = 2.0 * aold * bold + y; zmagsqr = a * a + b*b; aold = a; bold = b;}

// Record number of iterations for pixel.matrix_r[c] = huetable[i];

Parallelize the Program

• Each pixel value can be computed independently, so divide the matrix rows among the threads

• All inputs are shared

• No per-thread or local variables need synchronization

• No padding needed

The Parallel for Loop// Compute all rows and columns.new ParallelTeam().execute(new ParallelRegion(){ public void run() throws Exception{ execute (0, height-1, new IntegerForLoop(){ public void run (int first, int last){

for (int r = first; r <= last; ++ r){ int[] matrix_r = matrix[r];

double y = ycenter + (yoffset - r) / resolution;

Behavior of Parallel Program